Shock-free supersonic elliptic nozzles and method of forming same

Abstract

A method of forming a shock-free supersonic elliptic nozzle, in which the nozzle to be designed is divided into three sections, a circular-to-elliptic section which begins at a circular nozzle inlet, an elliptic subsonic section downstream from the circular-to-elliptic section and a supersonic section downstream from the elliptic subsonic section. The maximum and minimum radii for each axial point in the circular-to-elliptic section and the elliptic subsonic section are then separately determined, the maximum and minimum radii being the radii for the widest part of an elliptic cross-section and the narrowest part of the elliptic cross-section, respectively. The maximum and minimum radii for each axial point in the supersonic section are determined based on the Method of Characteristics, Then, each of the three sections are based on the maximum and minimum radii for each axial point in the section. The resulting nozzle is acoustically superior.

What is claimed is: 
     
       1. A method of forming an elliptic supersonic nozzle having unequal major and minor axes, comprising the steps of: (a) providing design coordinates of a round supersonic nozzle which will translate to the elliptical supersonic nozzle to be formed;   (b) translating the coordinates of the round nozzle into elliptic nozzle coordinates; and   (c) forming the elliptic supersonic nozzle based on the elliptic nozzle coordinates.   
     
     
       2. A method of forming an elliptic supersonic nozzle having unequal major and minor axes, comprising the steps of: (a) dividing the nozzle to be formed into three sections, a circular-to-elliptic section which begins at a circular nozzle inlet, an elliptic subsonic section downstream from the circular-to-elliptic section and a supersonic section downstream from the elliptic subsonic section;   (b) determining the major and minor radii for each axial point in the circular-to-elliptic section, the major and minor radii being the radii for the widest part of an elliptic cross-section and the narrowest part of the elliptic cross-section, respectively;   (c) determining the major and minor radii for each axial point in the elliptic subsonic section;   (d) determining the major and minor radii for each axial point in the supersonic section; and   (e) forming each of the three sections based on the major and minor radii for each axial point in the respective section.   
     
     
       3. The method of forming an elliptic supersonic nozzle as claimed in claim 2, wherein step (d) further comprises the substeps of: selecting an aspect ratio A r  greater than 1.0, the aspect ratio being the ratio of the major radius to the minor radius for each axial point in the elliptic subsonic section and the supersonic section;   determining the radii R(x) for each point of a supersonic section of a circular nozzle;   determining the major radii a(x) for each axial point of the supersonic section from the equation ##EQU6## determining the minor radii b(x) for each axial point of the supersonic section from the equation ##EQU7##   
     
     
       4. The method of forming an elliptic supersonic nozzle as claimed in claim 2, wherein step (c) further comprises the substeps of: selecting an aspect ratio A, greater than 1.0, the aspect ratio being the ratio between major radius and the minor radius for each axial point in the elliptic subsonic section and the supersonic section; determining the radii R(x) for each point of a contracting subsonic section of a circular nozzle;   determining the major radii a(x) for each axial point of the elliptic contracting section from the equation ##EQU8## determining the minor radii b(x) for each axial point of the elliptic contracting section from the equation ##EQU9##   
     
     
       5. The method of forming an elliptic supersonic nozzle as claimed in claim 4, wherein step (d) further comprises the substeps of: determining the radii R(x) for each point of a supersonic throat section of a supersonic circular nozzle;   determining the major radii a(x) for each axial point of the supersonic section from the equation ##EQU10## determining the minor radii b(x) for each axial point of the supersonic section from the equation ##EQU11##   
     
     
       6. The method of forming an elliptic supersonic nozzle as claimed in claim 4, wherein the aspect ratio is selected to be approximately 2. 
     
     
       7. The method of forming an elliptic supersonic nozzle as claimed in claim 3, wherein the aspect ratio is selected to be approximately 2. 
     
     
       8. The method of forming an elliptic supersonic nozzle as claimed in claim 2, wherein step (b) further comprises the substeps of: selecting a major nozzle radius A L , at a downstream end of the circular-to-elliptic section which is less than the circular nozzle inlet radius R i  and which results in a subsonic velocity at a downstream end of the circular-to-elliptic section; selecting a length X L  of the circular-to-elliptic section which will prevent gas separation at the wall of the circular-to-elliptic section;   selecting an aspect ratio A r  greater than 1.0, the aspect ratio being the ratio of the major radius to the minor radius for each axial point in the elliptic subsonic section and the supersonic section;   determining the major radii a(x) for each axial point of the circular to elliptic section from the equation   a(x)=R.sub.i +Sx+C.sub.1 x.sup.2 +C.sub.2 x.sup.3,         (1)       where S is the slope of the walls at the circular nozzle inlet, x is the axial distance from the circular nozzle inlet and C 1  and C 2  are constants;   determining C 1  from the equation ##EQU12## determining C 2  from the equation ##EQU13## determining the minor radii b(x) for each axial point of the circular to elliptic section from the equation ##EQU14##   
     
     
       9. The method of forming an elliptic supersonic nozzle as claimed in claim 8, wherein step (d) further comprises the substeps of: determining the radii R(x) for each point of a supersonic throat section of a supersonic circular nozzle;   determining the major radii a(x) for each axial point of the supersonic section from the equation ##EQU15## determining the minor radii b(x) for each axial point of the supersonic section from the equation ##EQU16##   
     
     
       10. The method of forming an elliptic supersonic nozzle as claimed in claim 8, wherein step (c) further comprises the substeps of: determining the radii R(x) for each point of a contracting section of a circular nozzle;   determining the major radii a(x) for each axial point of the elliptic contracting section from the equation ##EQU17## determining the minor radii b(x) for each axial point of the elliptic contracting section from the equation ##EQU18##   
     
     
       11. The method of forming an elliptic supersonic nozzle as claimed in claim 8, wherein the major nozzle radius A L  at a downstream end of the circular-to-elliptic section is selected so that a flow velocity of approximately Mach 0.3 results at the downstream end of the circular-to-elliptic section. 
     
     
       12. The method of forming an elliptic supersonic nozzle as claimed in claim 8, wherein the aspect ratio is selected to be approximately 2. 
     
     
       13. The method of forming an elliptic supersonic nozzle as claimed in claim 10, wherein step (d) further comprises the substeps of: determining the radii R(x) for each point of a supersonic throat section of a circular nozzle;   determining the major radii a(x) for each axial point of the supersonic section from the equation ##EQU19## determining the minor radii b(x) for each axial point of the supersonic section from the equation ##EQU20##   
     
     
       14. An elliptic jet nozzle, comprising: a circular-to-elliptic section attached at an upstream end to a circular inlet; an elliptic subsonic section having unequal major and minor axes downstream from the circular-to-elliptic section, the elliptic subsonic section having an aspect ratio which is constant for the section and a cross-sectional area which decreases in the downstream direction so that gases leaving the elliptic subsonic section are sonic as they leave; and   a supersonic elliptic section in which the gases flowing therein move at supersonic speeds, the supersonic section being downstream from the elliptic subsonic contraction section, having an aspect ratio which is constant for the section and an exit which allows supersonic gases to escape to the atmosphere.   
     
     
       15. The elliptic jet nozzle as claimed in claim 14, wherein the aspect ratio for the elliptic subsonic section is equivalent to the aspect ratio of the supersonic elliptic section.